Perpetuities in WACE Mathematics Applications Unit 4

Syllabus dot point

“Model a perpetuity, find the payment that keeps the balance constant, and relate it to the interest earned each period.”

WACEMathematics ApplicationsUnit 47 min readUpdated 2 Jun 2026

Quick answer

A perpetuity pays a fixed amount at the end of every period forever, by withdrawing exactly the interest earned each period so the principal never changes. The sustainable payment is $d = P i$ (principal times the per-period rate), and the principal needed for a target payment is $P = \dfrac{d}{i}$ . Because the withdrawal equals the interest, the balance returns to its starting value after every payment.

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What this dot point is asking
What a perpetuity is
The perpetuity formulas
Perpetuity versus draw-down annuity
Indexed (growing) perpetuities
Worked check of sustainability
Exam-style questions

What this dot point is asking

You must model a perpetuity, find the sustainable payment or required principal, and explain why it never depletes, contrasting it with a draw-down annuity.

What a perpetuity is

A perpetuity is an investment that pays a constant amount each period indefinitely. The trick is that the payment exactly matches the interest earned, so the principal is untouched.

Perpetuity

A perpetuity withdraws only the interest earned each period, leaving the principal unchanged. The balance is constant after every payment, so payments can continue forever. It is the boundary case of an annuity where the withdrawal equals the interest.

In recurrence terms, the perpetuity is $A_{n+1} = (1 + i)A_n - d$ with $d = i A_0$ , which simplifies so that $A_{n+1} = A_0$ for every $n$ : the balance never moves.

The perpetuity formulas

Perpetuity relations

Let $P$ be the principal, $i$ the per-period interest rate, and $d$ the regular payment.

Sustainable payment: $d = P i$ (one period's interest).
Principal for a target payment: $P = \dfrac{d}{i}$ .

Always convert the nominal annual rate to the per-period rate first (divide by periods per year), and match the payment to that period.

Withdraw only the interest

A perpetuity works because the withdrawal equals the interest the fund earns each period. Withdraw less and the fund grows; withdraw more and it becomes a depleting annuity. The exact match keeps the balance constant forever.

Worked example

Setting up a perpetuity

A trust of $\$400\,000$ earns $4.8\%$ per annum compounding monthly and pays an equal amount to a beneficiary at the end of each month forever. Find the monthly payment and verify the balance is constant.

Step 1: Convert the annual rate to the per-period rate.

Because the interest compounds monthly and payments are made monthly, we need the rate per month, not per year. Monthly rate $i = \dfrac{0.048}{12} = 0.004$ . Always match the rate to the payment period before applying any perpetuity formula.

Step 2: Find the sustainable monthly payment.

For a perpetuity the payment equals exactly one period's interest, so the principal is untouched after each payment. Applying $d = Pi$ : $d = 400000 \times 0.004 = 1600$ , so the payment is $\$1600$ per month.

Step 3: Verify that the balance stays constant.

We check that withdrawing $\$1600$ returns the fund to its starting value. At the start of the month the fund holds $\$400\,000$ . Interest earned during the month: $400000 \times 0.004 = \$1600$ , which raises the balance to $\$401\,600$ . Paying out $\$1600$ brings the balance back to exactly $\$400\,000$ . Because the withdrawal matches the interest earned, the fund returns to its starting value after every payment, confirming the $\$1600$ can be paid forever.

Step 4: Find the principal required for a larger payment.

If the beneficiary needs $\$2000$ per month instead, we rearrange to $P = \dfrac{d}{i}$ . The required principal is $P = \dfrac{2000}{0.004} = 500000$ , that is $\$500\,000$ . A larger payment demands a proportionally larger principal, because the fund must earn the full payment as interest each month.

Perpetuity versus draw-down annuity

The two differ only in the size of the withdrawal relative to the interest.

Perpetuity: withdrawal $=$ interest. Balance constant, lasts forever.
Draw-down annuity: withdrawal $>$ interest. Balance falls, fund eventually exhausts.

This is why a perpetuity question asks for the payment that "keeps the balance the same" or "can be paid indefinitely", whereas an annuity question asks "how long does the fund last".

Indexed (growing) perpetuities

Some perpetuities increase the payment over time to keep pace with inflation, while still never touching the original capital in real terms. In the SCSA course you handle the level (constant-payment) perpetuity, but it helps to see the boundary clearly: as long as the withdrawal does not exceed the interest earned, the fund survives. If a fund earns more interest than it pays out, the balance grows and the extra can be reinvested; if it pays out exactly the interest, it is a perpetuity; if it pays out more, it is a depleting annuity. Classifying a fund into one of these three cases by comparing one period's interest with the withdrawal is a frequently examined judgement.

Worked check of sustainability

Suppose a fund of $\$150\,000$ earns $4.2\%$ per annum compounding monthly and pays $\$600$ per month. The monthly interest is $150000 \times \dfrac{0.042}{12} = 150000 \times 0.0035 = \$525$ . Because the $\$600$ withdrawal exceeds the $\$525$ interest, the fund is not a perpetuity; it will slowly deplete. To make it a perpetuity the payment must be reduced to $\$525$ , or the principal raised to $P = \dfrac{600}{0.0035} = \$171\,428.57$ . This kind of "is it sustainable, and if not, fix it" question is the most common perpetuity task.

Common errors

Using the annual rate for a monthly payment: Convert to the per-period rate before applying $d = Pi$ .
Confusing a perpetuity with a draw-down annuity: A perpetuity withdraws only the interest, so it never depletes; an annuity withdraws more and runs out.
Inverting the principal formula: The required principal is $P = \dfrac{d}{i}$ , not $d \times i$ .
Forgetting to verify the constant balance: A clear answer shows the interest earned equals the withdrawal, returning the balance to its start.

Exam-style questions

Questions in the style of SCSA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

WACE 2022-style5 marks

A scholarship fund of

\$250\,000

is invested at

5.4\%

per annum compounding monthly. It is to pay an equal scholarship at the end of each month forever. (a) Find the monthly payment that keeps the balance constant. (b) Explain why the balance never falls.

Show worked answer →

A perpetuity withdraws exactly the interest earned each period.

(a) Monthly rate $= \dfrac{0.054}{12} = 0.0045$ . The sustainable payment equals one period's interest: $250000 \times 0.0045 = 1125$ , so $\$1125$ per month. (3 marks)

(b) Each month the fund earns $\$1125$ in interest and pays out exactly $\$1125$ , so the balance returns to $\$250\,000$ after every payment and never falls. (2 marks)

Markers reward the per-period rate, the payment equal to one period's interest, and the constant-balance reasoning.

WACE 2024-style5 marks

A charity wants to pay a grant of

\$3000

at the end of each quarter forever from a perpetuity earning

6\%

per annum compounding quarterly. (a) Find the principal required. (b) State how a perpetuity differs from a draw-down annuity.

Show worked answer →

The principal is the amount whose interest equals the payment.

(a) Quarterly rate $= \dfrac{0.06}{4} = 0.015$ . The payment equals one period's interest, so $P \times 0.015 = 3000$ , giving $P = \dfrac{3000}{0.015} = 200000$ , that is $\$200\,000$ . (3 marks)

(b) A perpetuity withdraws only the interest, so the balance stays constant and payments continue forever. A draw-down annuity withdraws more than the interest, so the balance falls and the fund eventually runs out. (2 marks)

Markers reward principal as payment divided by per-period rate and the never-runs-out distinction.